Global Journal of Mahemaical Sciences: Theory and Pracical. ISSN 0974-3200 Volume 9, Number 3 (2017), pp. 427 439 Inernaional Research Publicaion House hp://www.irphouse.com Pricing floaing srike lookback pu opion under heson sochasic volailiy Teferi Dereje Wiru Pan African Universiy Insiue for Basic Sciences, Technology and Innovaion (PAUSTI), Kenya Philip Ngare Deparmen of Mahemaics, Universiy of Nairobi, Kenya. Ananda Kube Deparmen of saisics, Kenyaa Universiy, Nairobi, Kenya. Absrac The pricing problems of he exoic opions in he finance do no have he analyic soluions under sochasic volailiy and so i is difficul o calculae he opion prices or a leas i requires much of ime o compue hem. This sudy provides he required heoreical framework o praciioners for he opion price esimaion. This paper focuses on pricing for floaing srike lookback pu opion and esing opion pricing formulas for he Heson sochasic volailiy model, which defines he asse volailiy as he sochasic process. Our pricing mehod is depending on he PDE approach on Heson sochasic volailiy model and homoopy analysis mehod. Heson model has received he mos aenion hen i can give a accepable explanaion of he underlying asse dynamics. The resuling formula is well conneced o a Black Scholes price ha is he firs erm of he series expansion, which makes compuing he opion prices fairly efficien. AMS subjec classificaion: Keywords: lookback pu opion, opion pricing,sochasic volailiy model, Heson model, homoopy analysis mehod.
428 Teferi Dereje Wiru, e al. 1. Inroducion European Look back opions are kind of he exoic opion wih pah-dependen, inroduced a firs by Goldman, Sossin and Gao (1997) having heir selemen based on he minimum or he maximum value underlying asse regisered during he life of he opion ime. A mauriy, he holder can look back and selec he mos convenien price of he underlying ha occurred during his period: herefore hey offer invesors he opporuniy a a price of buying a sock a is lowes price and selling a sock a is highes price. Since his scheme guaranees he bes possible resul for he opion holder, he or she will never regre he opion payoff. As a consequence, a look back opion is more expensive han any oher opion wih similar payoff funcion sochasic volailiy is a saisical model used for o evaluae derivaive securiies, such as opions.sochasic volailiy reas he volailiy of he underlying asse as a random variable. This improves he accuracy of models and forecass.sochasic volailiy models for he pricing derivaive securiies have been developed as he exension o he original, consan volailiy model of he Black and Scholes (1973). Under he Black and Scholes model, closed-form pricing formulas for he coninuously moniored lookback opions were derived by Goldman and Conze and Viswanahan. Heynen and Ka(1979) derived he analyical formulas for discreely moniored lookback opions under he Black-Scholes seing and he corresponding formulas were furher generalized o he general Là vy seing by Agliardi. In he Black-Scholes model, he underlying asse price process is assumed o follow he geomeric Brownian process, in which he volailiy of he asse price is a consan. This assumpion used in opion price wih he Black-scholes formula is inconsisen wih he phenomena of real daa in some sudies, for example see [8],[22].under he assumpion consan volailiy, opion price can no affeced by he change in he price of he underlying asse bu in he sochasic volailiy opion price can affeced by he change in he price of he underlying asse. So ha by assuming ha he volailiy of he underlying price is sochasic process raher han a consan, i becomes possible o model derivaives more accuraely. The pricing of he lookback opions under he sochasic volailiy model poses ineresing mahemaical challenges.in his work, we use he homoopy analysis mehod o derive he analyic pricing formulas for lookback opions under Heson s sochasic volailiy model There are a number of mehods one can use o model volailiy sochasically. Hull and Whie (1987) model he variance using a geomeric Brownian moion, as well as an Ornsein-Uhlenbeck process wih mean-reversion relaed o he volailiy. In he general case, mean-reversion is considered o be an essenial feaure of observed volailiy, and hus all plausible models are of he Ornsein-Uhlenbeck ype. Wiggins (1987) models he logarihm of he volailiy wih mean-reversion, whereas Sco (1987), Johnson and Shanno (1987), Heson (1993) and Sein and Sein (1991) model he variance using a square roo process. Zhu (2000) also considers a double square roo process, which is an exension of he basic square roo process in which boh he drif and diffusion coefficiens involve he volailiy. In his paper we focus on Heson⣠s square roo model,under which Heson (1993) provides an analyic expression for European opion prices.
Pricing floaing srike lookback pu opion under heson sochasic volailiy 429 This paper is concerned wih a pricing of he floaing srike lookback pu opion. The price of floaing srike lookback pu opion depends on he maximum or minimum of he underlying asse price. In his work, we exercise he homoopy analysis mehod o drive he pricing formulas for he floaing srike lookback pu opion under Heson sochasic volailiy model.the homoopy analysis mehod has been used by Liao (1992, 1997, 2003) in mechanics, and Zhu (2006), Zhu e al. (2010), Park and Kim (2011) and Leung (2013) in he finance.now we are going o explore he relaionship beween he lookback pu opion price wih he floaing srike and sock price. The paper is organized as follows. In Secion 2, we inroduce Heson sochasic volailiy model. In secion 3, we derive a governing parial differenial equaion (PDE) for he lookback opions. In Secion 4, By he homoopy analysis mehod we used o derive a pricing formula for a floaing srike lookback pu opion. The simulaion resuls are presened in secion 5.The conclusions are presened in Secion 6. 2. The heson sochasic model The Heson SV model assumes he following sochasic process for he underlying asse price s under risk neural measure a he ime given as: and he variance follows he process: ds = r S d + S v dw 1 (1) dv = k(θ v )d + σ v dw 2 (2) dw 1 dw2 = pd To ake ino accoun leverage effec, Wiener sochasic processes W 1,W 2 should be correlaed dw 1 dw2 = pd. In which he correlaion coefficien is p. The sochasic model (2) for he variance is relaed o he square-roo process of Feller (1951) and Cox, Ingersoll and Ross (1985). For he square-roo process (2) he variance is always posiive and if 2k >σ 2 hen i canno reach zero. Noe ha he deerminisic par of process (2) is asympoically sable if k>0. Clearly, ha equilibrium poin is V = 0. The parameers used in he model are as follows: S is he price of he underlying asse a he ime K is he rae of he mean reversion r is he risk free ineres rae Θ is he long erm mean variance v is he variance a he ime σ is he volailiy of he variance process.
430 Teferi Dereje Wiru, e al. Therefore, under he Heson model, he underlying asse follows an evoluion process which is similar o he Black-scholes model, bu i also inroduces a sochasic behavior for he volailiy process. In paricular, Heson makes he assumpion ha he asse variance V follows a mean revering Cox-Ingersoll-Ross process.consequenly, he Heson model provides a versaile modelling framework ha can accommodae many of he specific characerisics ha are ypically observed in he behavior of financial asses. In paricular, he parameer σ conrols he kurosis of he underlying asse reurn disribuion, while p ses is asymmery. We plo an example of he sock price and volailiy of he sochasic process in he Figure 1 and Figure 2 respecively. Figure 1: Sock price are dynamics in Heson model Figure 2: Volailiy dynamics in he Heson model
Pricing floaing srike lookback pu opion under heson sochasic volailiy 431 3. The PDE of he heson model for he opion price In his secion we derive he PDE ha he price of a derivaive mus solve, where he radeable securiy as well as he volailiy of he radeable securiy follows general sochasic processes. The PDE ha governs he prices of derivaives wrien on a radeable securiy wih sochasic volailiy is derived and we describe how o derive he PDE for he Heson model. This derivaion is a special case of a PDE for general sochasic volailiy models which is described by Gaheral. Heson is one of he mos popular opion pricing models. This is due in par o he fac ha he Heson model produces call or pu prices ha are in closed form, up o an inegral ha mus evaluaed numerically. In his noe we presen a complee derivaion of he Heson model. In order o price opions in a sochasic volailiy model, we can apply no-arbirage argumens, or use he risk-neural valuaion mehod. To derive heson PDE le Form a porfolio consising of one opion being priced,denoed by he value V = V (s, v, ), unis of he sock S, ψ of anoher opion U = U(s, v, ) ha is used o hedge he volailiy. The porfolio has value = V + S + ψu (3) where =. Assuming he porfolio is self financing, he change in porfolio value is d = dv + ds + ψu (4) Apply ios Lemma o dv and differeniae wih respec o he variables,s,v we ge dv = V V V d + ds + S v dv + 1 2 vs2 2 V S 2 d + 1 2 σ 2 v 2 V v 2 d + σvp 2 V d (5) v S Applying Ios Lemma o du produces he idenical resul, bu in U. Combining hese wo expressions, we can wrie he change in porfolio value as: d = dv + ds + ψu (6) { V d = + 1 2 vs2 2 V S 2 + pσvs 2 V v S + 1 } 2 vσ2 2 V v 2 d { + ψ + 1 2 vs2 2 U S 2 + pσvs 2 U v S + 1 } 2 σ 2 v 2 U v 2 d (7) { V + S + ψ } { V S + ds + v + ψ } dv v In order for he porfolio o be hedged agains movemens in he sock and agains volailiy, he las wo erms in Equaion (6) involving ds and dv mus be zero. This implies ha he hedge parameers mus be ψ = V v v, = ψ S V S (8)
432 Teferi Dereje Wiru, e al. Moreover, he porfolio mus earn he risk free rae,r.hence d = r d. Now wih he values of and ψ from Equaion (7) he change in value of he riskless porfolio is { V d = + 1 2 vs2 2 V S 2 + pσvs 2 V v S + 1 } 2 vσ2 2 V v 2 d { +ψ + 1 2 vs2 2 U S 2 + pσvs 2 U v S + 1 } (9) 2 σ 2 v 2 U v 2 d In risk less porfolio we have which we wrie as d = X + ψy X + ψy = r(v + S + ψu) Subsiuing for ψ and re-arranging, produces he equaliy X rv + rs V S V v d = r d (10) = Y ru + rs S v The lef-hand side of Equaion (11) is a funcion of V only, and he righ-hand side is a funcion of U only. This implies ha boh sides can be wrien as a funcion f(s, v, ) of S, v, and. Following Heson, specify his funcion as f(s,v,)= K(θ v)+ λ(,s,v) where λ(,s,v) is marke price of volailiy risk. Subsiue f(s v,) ino he lef-hand side of Equaion (11), subsiue for Y, and rearrange o produce he Heson PDE for he opion U expressed in erms of he price S + 1 2 vs2 2 U S 2 + pσvs 2 U v S + 1 2 vσ2 2 U v 2 ru +rs S +[K(θ v) λ(,s,v)] v = 0 Nex we can derive a closed form of a PDE in erms log price,le x = lns and describe he PDE in erms of x,v and insead of S,v, and.this leads o a simpler form of he PDE. We need he following derivaives,which are direc o derive s, 2 U v s, 2 U s 2 Inser ino he Heson PDE Equaion (12). All he S erms eliminaed and we obain he Heson PDE in erms of he log price x = lns and Heson assumes ha he price are risk-neural.the reason for his erm is ha in realiy mos invesors are found o be risk averse in experimenal seings [1]. Moreover, Lamoureux and Lasrapes find evidence from observed opion prices ha he efficien-marke hypohesis and invesor (11) (12)
Pricing floaing srike lookback pu opion under heson sochasic volailiy 433 risk-neuraliy canno hold simulaneously [2]. Ofen λ is assumed zero, however, Then he price is said o be given under he risk-neural measure, i.e, under he assumpion ha invesors are risk-neural. In he following formulaions we will assume λ = 0. + 1 2 v 2 U x 2 + pσv 2 U v x + 1 2 vσ2 2 U (r v 2 ru + v ) 2 x +[K(θ v)] v = 0 The parial differenial Equaion (13) shows ha i is a governing he opion price. 4. Floaing srike lookback pu opion price The payoffs of he floaing srike lookback pu opions depend on he maximum asse price reached during he life of he opion and underlying asse price observed a he mauriy. Based upon he fundamenal heorem of asse pricing (Shreve (2000)), he no-arbirage price of a European lookback pu opion wih floaing srike is given by (13) U f (,x,x,v)= E Q [e r(t ) H f (X T,X T ) X = x,x = x,v = v] (14) where XT is he maximum asse price observed during he life of he opion and H f is he payoff of he pu opion.in his secion we consider a floaing srike lookback pu opion, where he underlying asse price is assumed o follow he SDE.Then H f (X T,XT ) = XT S T (payoff pu opion), he risk-neural price of he floaing srike lookback pu opion, denoed by U f (,x,x,v) a he ime, [o, T ] for X = x,x = x and V = v is given as U f (,x,x,v)= E Q [e r(t ) H f (X T X T ) X = x,x = x,v = v] (15) In his secion H f = (XT X T ) floaing srike chosen.transforming he governing equaion (13) in erms of he differenial operaors explained by 1 = pσv 2 + pσv + k( v) v x v v + 1 2 σ 2 v 2 v 2 2 = + 1 2 v 2 (r x 2 + v ) 2 x and he inegral problem (15) can be ransformed ino he PDE problem as follows.since boh x and x are coninuous and non decreasing.so ha he quadraic variance and covariance of he [x,x ] and [x,x ] saisfy he following condiions and [x,x ]= lim 0 Σm i=0 ([x i+1 x i ])2 x lim 0 max(x i+1 x i ) = 0 [x,x ]= lim 0 Σm i=0 (x i+1 x i ) 2 (x,x i+1 x,x i )
434 Teferi Dereje Wiru, e al. lim max(x i+1 x i ) = 0 0 for some pariion ={0 = 0 1 2 =}. This gives he inegral involved wih he dx dx and dx dx can be zero. so from ios formula we can ge Then we have d(e r U f ) = e r ( 1 + 2 )U f d + U f x dx E[ T e rs ( 1 + 2 )U f ds + T U f x dx s x = x,x = x ] for,t [0, ] since e r U f is a maringale and he second par of he condiional expecaion is zero on he 0 <x<x so ha he PDE for he U f can obained on he inerval 0 <x<x and by using mean value heorem and aking he T we can ge ( 1 + 2 )U f (,x,x,v)= 0, 0 T,0 x x U(T,x,x,v)= x x x (,x,x,v) x=x = 0 Here, he final condiion follows from he definiion (16) direcly and he assumpion on he coninuiy of parial derivaives leads o he boundary condiion. Definiion: zeroh-order deformaion equaion. Le p [0, 1]denoe he embedding parameer and U 0 (,x,x,v)be he iniial approximaion of he U(,x,x,v)such ha as p increases from 0 o 1,U(,x,x,v)varies coninuously from he iniial approximaion U 0 (,x,x,v),such kind of he coninuous variaion or deformaions are defined by he zero order deformaion equaion. Applying he definiion and Following he same vein as Park and Kim [4] mehod, he homoopy analysis mehod is o solve U(,x,x,v)from (16) we can consruc a homoopy of he of (16).To consruc le us consider U(,x,x,v,p)denoing he soluion of a PDE problem given by H(,x,x,v,p)is equal o zero wih he final and boundary condiion of (16), where H, called a homoopy, is defined by (16) H(,x,x,v,p)= (1 p)( 2 U(,x,x,v,p) 2 U 0 (,x,x,v)) +p( 1 + 2 )U(, x, x,v,p),p [0, 1] (17) Here U 0 (,x,x,v)is he iniial value approximaion from Black-scholes formula for he lookback pu opion price wih he consan volailiy.the Black⣠Scholes formula is well-known and, for insance, see Wilmo (2006). By his choice of U 0, he homoopy problem becomes H(,x,x,v,p)= 2 U(,x,x,v,p)+ p 1 U(,x,x,v,p)= 0 (18)
Pricing floaing srike lookback pu opion under heson sochasic volailiy 435 wih he final and he boundary condiion of (16) we can apply he homoopy analysis mehod by he considering a Taylor series U(,x,x,v,p)= U n (,x,x,v,)p n, (19) n=0 where U n denoe a Taylor coefficien. Noe ha floaing srike lookback pu opion price U f is hen given by U f (,x,x,v,)= lim p 1 U(,x,x,v,p)= U n (,x,x,v) (20) Insering equaion (19) ino (18) and using a sandard perurbaion argumen, we obain formally a hierarchy of PDE problem as follows we ge n=0 2 U n (,x,x,v)+ 1 U n 1 (,x,x,v)= 0, U n (T,x,x,v)= 0, x (,x,x,v) x=x = 0 for all n = 1, 2, 3,... To find he soluion of he equaion (21) we use wo lemmas: i.e. a lemma abou a Feynman-kac formula for floaing srike lookback pu opion price and lemma abou he join probabiliy densiy of he wo Gaussian processes.for he convenience, we use he noaion E x,x [.] :=E Q [. S = x,s = x ] wheres and S are he soluion given by respecively, for some V R +. S = rs d + V S dw S = max u S u Lemma 4.1. If Z(, x, x,v) U 1,2 b (R + R 3 ) and solve he PDE problem and also U 1,2 b (R + R 3 ) is he funcion space of bounded funcions coninuously differeniable wih respec o >0and wice coninuously differeniable wih respec o (,x,x,v) R 3. 2 Z(, x, x,v)= g(, x, x,v),0 T,0 <x x, Z(T, x, x,v)= h(x, x ) Z x (,x,x,v) x=x = 0 (21)
436 Teferi Dereje Wiru, e al. where g and h saisfy he condiions g + h = o(e x2 +x ) as x and x hen T Z(, x, x,v)= E x,x [e r(t ) h(s T,ST ) e r( s) g(s, S s,ss, v)ds] Proof. see (7) heorem 2.2. Lemma 4.2. If H and H are he wo Gaussian processes defined by, H = (r 1 2 σ 2 ) + σw H = max{(r 1 2 σ 2 )s + σw s }, hen he join probabiliy densiy of a processes (H,H ) is given as r 1 2 σ 2 Q(H db,h 2(2c a) (r 1 dc) = σ 3 2 3 e σ 2 b 2 σ 2 ) 2 2 2σ 2 Proof. see (shreve 2000) heorem 7.2.1. (2c b)2 2σ 2 dbdc Using he above wo lemmas we ge he following resul on he soluion of he PDE problem (21) Theorem 4.3. Assume ha he floaing srike lookback pu opion price U f (,x,x,v) is represened as U f (,x,x,v)= Un f (,x,x,v), n=0 hen Un f (,x,x,v)is given by U f 0 (,x,x,v)= (1 + σ 2 ) ( xn δ + (T, x )) ( 2r x +e r(t ) x N δ (T, x )) x for n = 0 and U f n (,x,x,v)= σ 2 2r e r(t ) x T ln( x x ) ( x x ) 2r ( σ 2 N δ )) (T, x x x 2(2c b) 1 U n 1 (s, xe b,xe c,v) σ 3 (s ) 3 2 2 exp{r( s) + r 1 σ 2 σ 2 b (r σ 2 2 )2 2σ 2 (s ) (2c b)2 2σ 2 (s ) }dbdcds,
Pricing floaing srike lookback pu opion under heson sochasic volailiy 437 where 1 is given by equaion (16), for n 1, where N denoes he usual cumulaive normal disribuion, and v = σ. δ ± (, x) = 1 σ (lnx + (r ± 1 2 σ 2 )) Proof. Since U f 0 is he Black-scholes pu opion price.hus he PDE of (16) for U 1 saisfies he required condiions of lemma and also U n for n>1 are smooh o be U 2 o due o Q(H db,h U n (,x,x,v) = E x,x [ = E x,x [ = = = T T ln( x x ) T ln( x x ) T ln( x x ) dc) = o(e b2 c2 ), hen from boh lemmas we can obain e r( s) 1 U n 1 (s, S s,s s, v)ds] e r( s) 1 U n 1 (s, xe H s,xe H s, v)ds] T ( e r( s) 1 U n 1 (s, xe b,xe c, v)ds)q(h s db,hs dc) e r( s) 1 U n 1 (s, xe b,xe c, v)ds)q(h s db,h s dc)ds 2(2c b) 1 U n 1 (s, xe b,xe c,v) σ 3 (s ) 3 2 2 exp{r( s) + r 1 σ 2 σ 2 b (r σ 2 2 )2 2σ 2 (s ) (2c b)2 2σ 2 (s ) }dbdcds his proves he heorem. 5. Numerical simulaion and resul Firs, we sudy he performance of he pricing formula for floaing srike lookback pu opion price wih payoff (max T S S T ) in he heson sochasic volailiy model. We presen he numerical example of he homoopy approximaion resul on he floaing srike lookback pu opion price under a sochasic volailiy model of he Heson ype which is specified by he implemenaion is done by evaluaing he inegral of he heorem 4.1,and Mone-Carlo simulaion.the parameers r = 0.02, = 0.04.p = 0,v 0 = 0.04,k = 0.03,x = 110,x = 100,and =0.5 are used for he implemenaion. We observe in he opion values are very close beween our pricing formula which is a blue sars and Mone-Carlo simulaion which is a red line.
438 Teferi Dereje Wiru, e al. Figure 3: Price of he floaing srike lookback pu opions as a funcion of sock price in he Heson model. 6. Conclusion The homoopy analysis mehod used in his paper offers a simple analyic mehod for he pricing lookback pu opions under Heson sochasic volailiy model. The price is given by an infinie series whose value can be deermined once an iniial erm is given well. This paper uses he probabilisic argumen ogeher wih he semi-analyic mehod called he homoopy analysis mehod o obain an approximaion formula for he floaing srike lookback pu opion price under Heson sochasic volailiy model. The resulan formulas for he opion price are very close beween our pricing formula and mone-carlo simulaion. References [1] Holy, C.A. and Laury, S.K. (2002), Risk aversion and incenive effecs. American Economic Review, 92, PP. 1644 1655. [2] Lamoure, C.G. and Lasrapes, W.D., (1993), Forecasing sock-reurn variance: Toward an undersanding of sochasic implied volailiies, Review of Financial Sudies, 6, pp. 293 326. [3] Leung, K.S., (2013), An analyic pricing formula o lookback opions under sochasic volailiy, Applied Mahemaics vol. 26, pp. 145 149.
Pricing floaing srike lookback pu opion under heson sochasic volailiy 439 [4] Park, S. H., Kim, J.S., (2011), Homoopy analysis mehod for opion pricing under sochasic volailiy, Applied Mahemaics Leers 24, PP. 1740 1744. [5] Conze and Vishwanahan (2017), Pah-dependen opions: he case of look back opions. J. Finance, PP. 46, 1893 1907. [6] Black, F. and M. Scholes, (1973), European look back Opion pricing and Corporae Liabiliies, ž Journal of Poliical Economy 81, No.3, pp. 637 654. [7] Seven E. Shreve, (2004), Sochasic calculus for finance II: Coninuous-ime models, volume 11. Springer. [8] Heson, S. (1993), A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions, Review of Financial Sudies 6, PP. 327 343. [9] Sco, L., (1987), Opion Pricing when he Variance Changes Randomly: Theory, Esimaion, and an Applicaion The Journal of Financial and Quaniaive Analysis 22, PP. 419 438. [10] Sein, E., Sein J., (1991), Sock Price Disribuions wih Sochasic Volailiy: An Analyic Approach, The Review of Financial Sudies 4, PP. 727 752. [11] Hull, J. Whie, A., (1987), The Pricing of Opions on Asses wih Sochasic Volailiies, The Journal of Finance 42, PP. 281 300. [12] Black, F., Scholes, M., (1997), The pricing of opions and corporae liabiliies, Journal of Poliical Economy 81, PP. 637 654. [13] Sco, L., (1987), Opion Pricing When he Variance Changes Randomly: Theory, Esimaors, and Applicaions, Journal of Financial and Quaniaive Analysis, 22, PP. 419 438. [14] Zhu, J. (2000), Modular Pricing of Opions: An Applicaion of Fourier Analysis, Lecure Noes in Economics and Mahemaical Sysems: 493, Springer Verlag Berlin Heidelberg. [15] Goldman, B.M.H.B. Sosin and M. A. Gao, (1979), Pah dependen opions: Buy a he low, sell a he high, Journal of Finance, 34, PP. 1111 1127. [16] Gourieroux, C., Monfor, A., and Renaul, E., (1993), Indirec Inference Journal of Applied Economerics, Volume 8, pp. 85 118. [17] Liao, S.J., (1992), The proposed homoopy analysis echnique for he soluion of nonlinear problems, Shanghai Jiao Tong Universiy. [18] Zhu, S.-P., (2006), An exac and explici soluion for he valuaion of American pu opion, Quaniaive Finance 6, PP. 229 242. [19] Karazas, I. and Shreve S.E., (1991), Brownian Moion and Sochasic Calculus, Springer-Verlag. [20] Leung, K.S., (2013), An analyic pricing formula for lookback opions under sochasic volailiy, Applied Mahemaical Leers, PP. 145 149. [21] Royden, H.L., Fizparick, P.M., (2010), Real Analysis. Prenice Hall. [22] Shiryaev, A.N. (1998), Foundaion of Sochasically mahemaics. Fazis, Moscow.